I believe you are referring to the function $\lambda$ defined in the first paragraph of page $250$ which carries an extra $-1/n$ (this is irrelevant for the invariance). If I am not missing anything, you have the following.
Define $g_{\ell}\overset{\Delta}= \limsup_{k\rightarrow \infty} \frac{1}{k} \sum_{i=0}^{k-1} fT^{i+\ell}.$
Then, $g_{\ell}=g_0$ for all $\ell\in\mathbb{N}$ and thus, $\psi_n T=g_1 \wedge n = g_0 \wedge n = \psi_n.$